The general optimisation framework we have suggested recognises that in many complex trials we are interested in minimising more than one quantity subject to constraints on operating characteristics. Problems of this sort are common in multilevel trial design [1]}, but are typically approached by first reducing the multiple objectives down to a single objective. For example, in the design of a cluster randomised trial it is common to fix the number of participants per cluster and minimise the number of clusters [2]}, or vice-versa [3]}, [4]}. Alternatively, a function which specifies the cost of sampling at the cluster and the patient level could be specified [5]}, and the overall cost minimised [6]}. The latter approach has been suggested for both two-level [7]} and three-level hierarchical trial designs [8]}, [9]}. However, the a priori specification of such a cost function may not always be feasible, particularly when several stakeholders are involved [10]}. The Pareto optimisation framework we have described leads to a more computationally challenging optimisation problem, but produces a set of good solutions enabling the available trade-offs between objectives to be seen and selected from.
